generalized integrals

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21: 19.15 Advantages of Symmetry

… ►Symmetry makes possible the reduction theorems of §19.29(i), permitting remarkable compression of tables of integrals while generalizing the interval of integration. …

22: 1.17 Integral and Series Representations of the Dirac Delta

… ►From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that … ►More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. … ►Hence comparison with (1.17.5) shows that (1.17.9) can be interpreted as a generalized integral (1.17.3) with … ►The sum ${\sum }_{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}k(x-a)}$ does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that …

23: 19.20 Special Cases

… ►The general lemniscatic case is … ►The general lemniscatic case is …

24: 2.6 Distributional Methods

… ►

2.6.3 $${\int }_0^{\mathrm{\infty }}\frac{t^{-s-(1/3)}}{x+t}dt,$$ $s=1,2,3,\mathrm{\dots }$.

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Symbols:

$dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.6.E3

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Annotations for §2.6(i), §2.6 and Ch.2

►Although divergent, these integrals may be interpreted in a generalized sense. … ►Corresponding results for the generalized Stieltjes transform … ►

2.6.58 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda }dt,$$ $\lambda \in ℝ$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\int$: integral and $ℝ$: real line

Permalink:

http://dlmf.nist.gov/2.6.E58

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Annotations for §2.6(iv), §2.6 and Ch.2

►However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. …

25: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

35.8.12 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\gamma -\frac{1}{2}(m+1)}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-𝐗)d𝐗={\mathrm{\Gamma }}_m\left(\gamma \right){\left|𝐓\right|}^{-\gamma }{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\gamma }{b_1,\mathrm{\dots },b_q};-{𝐓}^{-1}),$$ $\mathrm{\Re }\left(\gamma \right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $p$: nonnegative integer, $q$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E12

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Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

►

Euler Integral

►

35.8.13 $\mathrm{\Re }\left(b_1-a_1\right),\mathrm{\Re }\left(a_1\right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $p$: nonnegative integer, $q$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.8.E13

Encodings:

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See also:

Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

►

§35.8(v) Mellin–Barnes Integrals

…

26: 2.11 Remainder Terms; Stokes Phenomenon

… ►From §8.19(i) the generalized exponential integral is given by ►

2.11.5 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}{z}^{p-1}}{\mathrm{\Gamma }\left(p\right)}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-zt}{t}^{p-1}}{1+t}dt$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\int$: integral

Referenced by:

§2.11(iii)

Permalink:

http://dlmf.nist.gov/2.11.E5

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Annotations for §2.11(ii), §2.11 and Ch.2

… ►

2.11.6 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm and $E_p\left(z\right)$: generalized exponential integral

Referenced by:

§2.11(ii), §2.11(iii), §2.11(iii), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E6

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Annotations for §2.11(ii), §2.11 and Ch.2

… ►However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … ►Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector . …

27: Bibliography O

… ►

Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral. SIAM J. Math. Anal. 22 (5), pp. 1460–1474.

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External Links:

ISSN 0036-1410, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§2.11(iii), §2.11(iii), §8.11(i), §8.20(i).

Encodings:

BibTeX

… ►

The Generalized Exponential Integral. In Approximation and Computation (West Lafayette, IN, 1993), R. V. M. Zahar (Ed.), International Series of Numerical Mathematics, Vol. 119, pp. 497–510.

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External Links:

MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§2.11(iii), §8.19(viii).

Encodings:

BibTeX

…

28: 6.4 Analytic Continuation

… ►The general value of $E_1\left(z\right)$ is given by ►

6.4.1 $$E_1\left(z\right)=\mathrm{Ein}\left(z\right)-\mathrm{Ln}z-\gamma ;$$

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Ein}\left(z\right)$: complementary exponential integral, $E_1\left(z\right)$: exponential integral, $\mathrm{Ln}z$: general logarithm function and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E1

Encodings:

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See also:

Annotations for §6.4 and Ch.6

…

29: 22.14 Integrals

… ►

22.14.2 $$\int \mathrm{cn}(x,k)dx=k^{-1}\mathrm{Arccos}\left(\mathrm{dn}(x,k)\right),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real and $k$: modulus

A&S Ref:

16.24.2

Permalink:

http://dlmf.nist.gov/22.14.E2

Encodings:

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Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.3 $$\int \mathrm{dn}(x,k)dx=\mathrm{Arcsin}\left(\mathrm{sn}(x,k)\right)=\mathrm{am}(x,k).$$

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Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real and $k$: modulus

A&S Ref:

16.24.3

Permalink:

http://dlmf.nist.gov/22.14.E3

Encodings:

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Annotations for §22.14(i), §22.14 and Ch.22

… ►

22.14.5 $$\int \mathrm{sd}(x,k)dx={(k{k}^{\prime })}^{-1}\mathrm{Arcsin}\left(-k\mathrm{cd}(x,k)\right),$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{sd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arcsin}z$: general arcsine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.5

Permalink:

http://dlmf.nist.gov/22.14.E5

Encodings:

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See also:

Annotations for §22.14(i), §22.14 and Ch.22

►

22.14.6 $$\int \mathrm{nd}(x,k)dx=k_{}^{\prime }{}_{}{}^{-1}\mathrm{Arccos}\left(\mathrm{cd}(x,k)\right).$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{Arccos}z$: general arccosine function, $x$: real, $k$: modulus and $k^{\prime }$: complementary modulus

A&S Ref:

16.24.6

Permalink:

http://dlmf.nist.gov/22.14.E6

Encodings:

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See also:

Annotations for §22.14(i), §22.14 and Ch.22

…

30: 13.10 Integrals

… ►

13.10.3 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}𝐌(a,c,kt)dt=\mathrm{\Gamma }\left(b\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;c;k/z),$$ $\mathrm{\Re }b>0$, $\mathrm{\Re }z>\max (\mathrm{\Re }k,0)$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/13.10.E3

Encodings:

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See also:

Annotations for §13.10(ii), §13.10 and Ch.13

… ►

13.10.7 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{b-1}U(a,c,t)dt=\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(b-c+1\right)z^{-b}{}_2{}^{}𝐅_1^{}(a,b;a+b-c+1;1-\frac{1}{z}),$$ $\mathrm{\Re }b>\max (\mathrm{\Re }c-1,0)$, $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.10.E7

Encodings:

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See also:

Annotations for §13.10(ii), §13.10 and Ch.13

… ►Generalized orthogonality integrals (33.14.13) and (33.14.15) can be expressed in terms of Kummer functions via the definitions in that section.